Promoting, advancing and defending Intelligent Design via data, logic and Intelligent Reasoning and exposing the alleged theory of evolution as the nonsense it is.
I also educate evotards about ID and the alleged theory of evolution one tard at a time and sometimes in groups

Wednesday, June 19, 2013

This Just In- The Stupidity Nevers Ends

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This just in:

The set of non-negative integers, {0,1,2,3,4,5,6,7,8,9,10,...} does NOT contain the set of non-negative even integers, {2,4,6,8,10,...} AND have the positive odd integers left unmatched.

For some reason it is NOT correct to use the naturally derived alignment, ie exact matching of numbers, and instead an artificially constructed, ie contrived, alignment, which transforms all members into generic elements, is preferred. Yet no one can why nor what good that does.

Cantor is very consistent. If you want to see if two sets have the same cardinality put then into a one-to-one correspondence. That's it. You just don't get it. You're all hung up on the way the set members are counted and their 'values'.

And you can't seem to ever deal with comparing the cardinalities of {1, 2, 3, 4 . . . } and {1, ½, ⅓, ¼ . . . }You're method is in trouble 'cause you think the rate you count the elements matters and counting the second set is a problem for you. Not for Cantor's method. Go figure.

keiths sez:1. Whether you realize it or not, you applied the mapping function F(n) = 1/n in order to establish a one-to-one correspondence between the two sets. You then concluded that their cardinalities are the same.

Just because you have been unable to grasp what I have been saying doesn't mean anything to me.

2. In other words, you applied Cantor’s method (finally!) and got the correct answer (finally!).

I applied my method and it is just a coincidence that Cantor and I came up with the same ansser.

3. On the other hand, JoeMath (and its patented choo-choo train method) were utterly unable to deal with Jerad’s problem.

No, socle, only in the magical world of infinities, even infinities between 2 finite points.

That said, for those of you who think that I am using Cantor's method, I offer this:

In Joemath, using the same method I used wrt {0,1,2,3,4,...} and {1,1/2,1/3,1/4,1/5,...}, The set of non-negative integers would have a cardinality that is greater than the set {1,1/2,1/4,1/8,1/16,...}.

If I was using Cantor's method the cardinalities would still be equal.

As I said it was only a coincidence that Cantor and I agree in the first scenario. And it is very telling that you morons are so stupid that you couldn't see that- even though I told you.

And by exactly the same reasoning, (0, 1] and [1, ∞) have the same cardinality. The elements in one set are exactly the reciprocals of the elements in the other set.

You wouldn't use the reciprocals. They are already on the same terms.

They both start at 1 and start counting from there. And in magical infinity math neither would get anywhere.

JoeMath wouldn't use the train method. JoeMath would say [1, ∞)'s cardinality > (0, 1], and the reasoning would be that [1, ∞) has infinitely more integers and therefor infinitly more points between those integers.

I've had enough of this. I'm fed up with your bullying and inability to understand basic, clear arguments and misunderstanding of your own techniques.

I think now that it's right not to argue with denialists, it just gives them the attention they crave and because they have ulterior motives for many of the positions they hold they won't back down or change their minds.

It's not about reason or trying to understand or find the truth. It's about something else.

I"ve been wrong many times in my life, just about every day I make a mistake of some kind. And I try to learn lessons from all those events. It's pointless to try and teach people who don't want to learn.

My response would then be, you should be able to use the idea of reciprocals---why not, if it's valid in Jerad's example?

In fact, I could write the sets like this:

A = {x : 0 < x <= 1}

B = {1/x : 0 < x <= 1}

Then it's clear that every element of B is the reciprocal of an element in A, and also the other way around.

There are often several ways to solve a particular problem, and of course they all should give you the same result.

JoeMath wouldn't use the train method. JoeMath would say [1, ∞)'s cardinality > (0, 1], and the reasoning would be that [1, ∞) has infinitely more integers and therefor infinitly more points between those integers.

While I don't agree with your final conclusion, if you accept that [1, ∞) has infinitely more integers than (0, 1], then that certainly means that {1, 2, 3, ...} is an infinite set, and I would be happy to consider that point as settled.

That's correct, at least in my view. Just like the game of chess is an invention. OTOH, I wouldn't be surprised if every reasonably advanced civilization had invented the real numbers; maybe not chess-like games though.

Perhaps a historical perspective will help: At first, Cantor was thinking about finite sets. How do we find out that finite sets have the same size? A shepherd may map his sheep to a bundle of tokens, making sure that each token represents exactly one particular sheep and vice versa. A mathematician may look at the sets {1,2,3} and {a,b,c} and see that there is a invertible one-to-one mapping between this sets: in fact, 3 out of the possible 27 mapping between these sets are injective and surjective (bijective). There is is no bijective mapping between - say - {apple, orange} and {1/2,1/3,1/4}. So, we may define that two finite sets have the same size if there is a bijective mapping between those two.

Cantor was a curious man. He just thought what happened when we take this definition to infinite sets. This lead to some statements which may appear paradoxical ("an infinite set is a set which has the same size as one of its proper subsets") and perhaps that is the reason that he used the expression "cardinality" instead of "size".

Nevertheless, it works - even if you don't like it. And the idea of "countable infinite" sets has proven very successful: you can't do analysis or probability theory without it (ever heard of a sigma-algebra?)

Bottom point: Cantor's ideas work for mathematicians. That's all we are caring about.

All one has to do is count the number of members in a finite set- one doesn't have to know anything about mapping- onr-to-one or not.

And what do you mean by "it works" wrt infinite sets?

No one even uses it for anything.

And how can one set contain all of the members of the other set PLUS have members the other set does not, meaning the first set obviously has more members/elements, and yet cantor sez they are the same size, ie have the same cardinality?

And how can one set contain all of the members of the other set PLUS have members the other set does not, meaning the first set obviously has more members/elements, and yet cantor sez they are the same size, ie have the same cardinality?

Strange that the cowards always ignore that fact.

That's the beauty: you don't even have to know how to count when you wish to compare the size of to flocks of sheep - you only have to pair them up!

You have to count to know they are paired.

The shepard should know how many are in his flock. To have to walk around to find another flock to pair them with sounds like your accepted methodology, but it is also pretty stupid.

And even then you wouldn't know how many you had!

We get a consistent theory.

In what way? As I have said it is inconsistent wrt comparing sets. We use one methodology for seeing if one set is a proper subset of another, and then use a totally different alignment for comparing cardinalities of infinite sets.

That is inconsistent. And you are being vague- another cowardly sign.

It is used the very moment you start calculus!

Stop being an asshole. Calculus does not need the premise that all countably infinite sets have the same cardinality.

No one needs it. It is not used for anything.

Get a grip.

Also deal with the bolded part. Failure to do so will just prove your cowardice. And cowardice is not a way to win an argument nor convince your opposition that you have a case.

And yes, I have heard of sigma-algebra. Do you have a point?

To me it looks like you jumped in here without a clue as to what is being debated.

Stop being an asshole. I can't remember even to have started to be one!

But to answer your question: And how can one set contain all of the members of the other set PLUS have members the other set does not, meaning the first set obviously has more members/elements, and yet cantor sez they are the same size, ie have the same cardinality?

That's because they are infinite sets: it's a property of infinite sets to be of the same cardinality as some of their proper subsets! You don't have to like this, but that's how math is done.

That has nothing to do with magic - it's about making some postulates and then thinking them through.

Take e.g. geometry: what happens when we don't say that there is always exactly one parallel through a point given a straight line, but many? We wouldn't expect it from our usual knowledge of the world, but it turns out that we can draw conclusions and get to the consistent theories of non-Euclidian geometries.

What happens when we declare the equation x^2-1=0 to be solvable and think of the properties of the solution? We get the imaginary numbers, again not obvious from daily life, but very useful.

The same for cardinalities: what happens when we declare that two sets have the same cardinality if there is a bijective mapping between these maps? It seems that we get very far!

Sequences play an important role in calculus. What happens to the limit of a convergent sequence if you change a finite number of it elements? What happens to the limit if you drop every second element? Nothing! Is it useful to think of the new sequence as having less elements than the original? Turns out, it isn't. That's why generations of undergraduates are confronted with the marvels of Hilbert's hotel: it is a strange place and it takes some time to understand its working, but it is fund to live there!

This isn't about right or wrong, this is about one working concept vs. another working concept.

Obvioulsy no one thought it through.

Every student starting mathematics has to try to think it through. If he doesn't like it, fine, he can start with his own set of axioms and definitions and look where he is lead...

Your idea has some similarities to Nonstandard Analysis (which is much harder to understand than the usual way to think about numbers...)

The main drawback in your system: you cannot compare most of the sets which are usually discussed.

What's bigger (and why) in size: {0,1,2} or {1,2,3}

{0,1,2,3,...} or {1,2,3,4,...}

{0,1,2,3,...} or {-1,1,2,3,...}

{1,2,3,...} of {-1,-2,-3,...}

Cantor's system allows for a consistent way to compare the cardinalities of sets. Again, if you learn standard analysis or a little bit of measure theory and advanced analysis, topology, etc., it is used extensively.